There is some advantage to a slow diode, besides cost and voltage drop.
The slow recovery is also gradual, dissipating reactive energy as heat rather than snapping off and creating a sharp or ringing waveform that can conduct/radiate electromagnetic emissions (radio interference). The "bug" can be a feature!
The surge rating is slightly higher, more or less reflecting the lower VF, though there may be other reasons.
A few words about standards:
1Nnnn are (mostly) ancient JEDEC standards. In this earliest type system, parts were manufactured to target multiple types, and after testing and packaging, the same chip (process and masks) might be sold as many types. Multiple manufacturers sourced these parts, as demanded by major customers at the time -- military especially. You could be assured that any make of these parts will meet the common standard.
As such, these standards were often very lax. Notice how many specs are limiting values: minimums or maximums. 2N2222 for example only specifies minimum fT. How did you know you weren't going to get some random hotrod RF part "mis"labeled as one? That's the neat part, you don't. Probably this didn't happen very often because such a chip can be sold as something more profitable, but it did affect other types. The (originally) ponderously slow 2N3055 is a prime example: when manufacturers discontinued the original hometaxial process and migrated these parts to modern epitaxial lines, the fT went up by several times. Amplifiers built with long wires to the transistors, that formerly were normal and stable, became power oscillators!
These limits affect the 1N540x series as well. You're guaranteed to get something that meets the standard -- but how much it exceeds it (or what the variance is on min/max constrained values), you don't know.
We know that rectifier diodes are heavily optimized for low forward drop.
It seems, with process improvements over the decades (mostly the earliest decades; this too is ancient history), the VF/VRRM ratio has been optimized well enough that meeting the basic spec is trivial.
Once the basic spec has been met, is there really any value in pushing further? You might assume "strictly better" means unconditionally better, but might there be more value in optimizing other parameters instead?
It turns out there is. In fact, an average 1N5408 likely breaks down over 1400V. It seems most manufacturers have decided that erring on the side of robustness is a higher priority than further reducing VF. Since mains surges are in the couple-kV range, this greatly increases reliability -- even when used without MOVs or other surge protection devices.
You would actually have to look at avalanche rectifiers to get that characteristic -- although even among them, you don't see it very often. Here's an example that does specify V(BR): Vishay BYW82. Even here, it seems likely they target the worst case type in the series; presumably avalanche current could be higher at lower V(BR) for applicable types, but they didn't design this standard that way. (I'm not sure of the history of the BYW and related series; it may be proprietary to a manufacturer, I don't know.)
Furthermore, notice how all parts in the family are specified with the same values. Interesting, huh? You may find a 1N5401 breaks down anywhere from 400 to 1800V! (I haven't seen any "standard" rectifiers below this range, myself. Checking: correction, have seen one 1N4001 at 363V. Well, close.)
So, for repair purposes, meet or exceed VF, VRRM, IF(AV), IFSM, and trr if it's a fast type. Substituting fast for GP, or substituting PN ↔ schottky, or excessive CJO, etc., may worsen EMI.
You could indeed substitute modern parts and likely improve efficiency, but a complete evaluation would have to be done to eke out every point of gain (e.g. adjusting or removing snubbers), and then EMI would have to be checked, and either filtering improved or compromises made to get it back under limits if exceeded.
And I emphasize EMC here, mainly because it's the least understood topic by EEs, and most likely to change with these kind of substitutions. In practical terms, it might matter very little. As far as I know, in most places you stand very little chance of being prosecuted by authorities, to whatever extent regional authorities have the right to do so in the first place. But it's still a practical matter, if nothing else, and a spooky one at that. Maybe you don't notice anything wrong, ever. Maybe you get a persistent buzzing sound in your audio equipment now. (Cheap audio equipment is especially sensitive to RF; the buzzing and blipping sounds of GSM telephone communications are likely familiar to many people.) Maybe there's increased noise in your lab environment (making ~mV level measurements on the oscilloscope suddenly awkward). Maybe your neighbor's garage door stops working. Etc. It's a spooky situation, it may seem arbitrary and random. And it can be hard to track down. EMI tends to conduct throughout its (wiring) environment, you might measure similar levels throughout the building. The cost is likely small (unlikely to be a problem, and when it is, it's likely more just an annoyance), but persistent and so it may add up over time; meanwhile, fixing it is no small feat, most people don't have the equipment to track down and evaluate such equipment. So it's best to avoid such issues in the first place.